^ftcg

NBS

PubliNBS TECHNICAL NOTE 1171

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

-QC

100

U5753

No. 1171

1932
NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

LIBRARY MAR 2 4 1983 An Electromagnetic Formulation for note «

&'

Treating Optical Reflections from

Graded-Material Surfaces n

"<««

A. George Lieberman

Center for Manufacturing Engineering

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

Q

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued December 1982
National Bureau of Standards Technical Note 1171

Natl. Bur. Stand. (U.S.), Tech. Note 1171, 36 pages (Dec. 1982)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1982

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

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(Add 25 percent for other than U.S. mailing)
CONTENTS

Page

1. INTRODUCTION , 2

2. REFLECTION COEFFICIENT FORMULATION FOR AN ARBITARY

ONE-DIMENSIONAL MATERIAL TRANSITION 4

2.1 Reflection Coefficient for an s-Polarized Plane Wave . . 6

2.2 Reflection Coefficient for a p-Polarized Plane Wave ... 14

2.3 Singularities of the Reflection Coefficient

Formulation for a Lossless Jellium Metal 17

3. WAVE IMMITTANCE FORMULATIONS FOR AN ARBITARY

ONE-DIMENSIONAL MATERIAL TRANSITION 21

3.1 Wave Immittance Formulations for s-Polarized

Plane Wave Reflections 21

3.2 Wave Immittance Formulations for p-Polarized

Plane Wave Reflections 25

3.3 Singularities of the Wave Immittance Formulations

for a Lossless Jellium Metal 25

4. CONCLUSIONS 29

5. REFERENCES 29

APPENDIX 31

in

AN ELECTROMAGNETIC FORMULATION FOR TREATING OPTICAL REFLECTIONS

FROM GRADED-MATERIAL SURFACES

A. George Lieberman

National Bureau of Standards

Washington, DC 20234

ABSTRACT

The reflection of a monochromatic plane wave falling obliquely upon the

surface of an arbitrary, flat, depth-dependent material is investigated

theoretically. The complex reflection coefficient for either principal

(s or p) polarization of the field is shown to satisfy a non-linear

differential equation of the Ricatti type. An alternate formulation

based on the wave immittance (i.e., jmpedance or a dmittance ) functions

is also presented. The immittance functions are shown to satisfy Ricatti

differential equations of their own. The reflection coefficient formulation

and the wave immittance formulations are related via a bilinear algebraic

transformation. Singularities appearing in the reflection coefficient

formulation may be suppressed in the immittance formulation, and viceversa. The advantage of either formulation is that the reflection

coefficients for an arbitrary, depth-dependent medium can be obtained

directly without having to solve Maxwell's equations for the internal

field configurations.

Key Words: electromagnetic waves; graded materials; inhomogeneous media;

jellium; optical reflections; reflection coefficient; Ricatti equation;

surface reflections; wave immittance.
1. INTRODUCTION

A general formulation is presented describing the field reflection

coefficients for an obliquely-incident monochromatic plane wave reflected

from the surface of a medium having arbitary depth-dependent properties.

The reflection coefficient for each principal linear polarization of the

incident field is shown to satisfy a nonlinear, ordinary differential

equation of the Ricatti type. The computation of a reflection coefficient

by this means avoids the sometimes impossibly difficult solution of Maxwell's

equations for the vector fields. The present method is significantly

advantageous for dealing with inhomogeneous media because the interior

fields are not simply configured except for a few special cases. Fortunately

the interior field configuration is rarely of interest and the present

formulation provides direct access to the plane wave reflection coefficients.

The methodology developed in this paper is rooted in a theoretical analysis

that Foersterling [1]1 provided in 1913 and 1914 to explain the Lippmann color

photographic process [2]. In those articles, and in a later one [3] treating

radio wave propagation within the Heaviside layer of the ionosphere,

Foersterling showed that the component of the local propagation vector

along the normal to the plane of stratification obeys a Ricatti differential

equation. Geffcken in 1941 [4] went on to show that for normal incidence

a characteristic function could be defined which also satisfied a Ricatti

differential equation and from which, outside the inhomogeneous medium,

the usual plane wave reflection coefficient could be recovered. Kofink

[5] later extended Geffcken' s work to include oblique incidence. The

1 Numbers in brackets indicate literature references listed at the end

of this report.
utility of these various formulations was, however, handicapped by the

non-linearity of the differential equations and the absence of today's

high speed computers. A number of analytical papers consequently followed

[6,7] concerned with approximation techniques which could be applied for

calculating, e.g., oblique reflections from thin ionospheric layers or

thick ionospheric layers with slowly varying parameters.

The work described above does not appear to be widely known outside

the radio physics community although some use of it has been made in the

study of non-uniform transmission lines [8,9] and acoustic ducts [10].

The present effort draws on this earlier work and restructures the

reflection formulation in a modern, more generally relevant manner. One

important aspect of the present formulation is that its application to

totally reflecting materials, such as metals, is now included in the

theory. An alternative formulation based on the wave impedance and wave

admittance functions is also presented. Each of these functions is

shown to obey its own Ricatti differential equation. The reflection

coefficient formulation and the wave immittance^ formulations are in

fact related by a bilinear algebraic transformation. In a given application the selection of a particular formulation is dictated by the

medium. The superiority of the wave immittance formulation over the

reflection coefficient formulation is demonstrated for a jellium-metal

surface in the final sections of this report.

2 Wave immittance refers to the impedance or a dmittance of the electromagnetic field in a plane parallel to the surface.
2. REFLECTION COEFFICIENT FORMULATION FOR AN ARBITRARY ONE-DIMENSIONAL

MATERIAL TRANSITION

Exact solutions of Maxwell's equations, in terms of well-known

functions, are available for only a few simple depth-dependent material

geometries. In those cases the complex reflection coefficient is determined by solving Maxwell's equations for the fields within the inhomogeneous

region, applying boundary conditions to match the interior fields with

the exterior plane wave fields, and finally forming the ratio of the

appropriate components of the incident and reflected plane wave fields.

This procedure may not be easy or practicable to perform. Fortunately,

the theory can be reformulated to provide the reflection coefficient

directly as the solution of an ordinary differential equation, and thereby

completely circumvent the field solutions. This approach is particularly

convenient when the material properties are numerically specified.

In the present formulation the entire space (-».<_ z <_ °°) is filled

with a medium whose properties are permitted to vary with coordinate z

(fig. 1). For sufficiently negative z the (exterior) medium is homogeneous

and supports the incident and reflected plane waves. In the vicinity of

z = the medium undergoes a transition which eventually develops, for

sufficiently positive z, into a second homogeneous region. The latter,

representing the uniform interior of the medium, supports the transmitted

plane wave. For the sake of generality both the permittivity (e) and the

permeability (y) are treated as arbitrary, independent functions which

for large positive and negative values of z approach constant values.

Losses are included by treating the permittivity and permeability functions

as complex entities in the frequency domain.

The reflection process commences with an incident plane wave, which
T7(z)

*

Incident Wave — Reflected Wave

Homogeneous Region

(Vacuum)

Inhomogeneous

(Surface)

Transition

Transmitted Wave

Homogeneous Region

(Medium) - z

Figure 1. A schematic representation of the medium depicting a possible

dependence of the refractive index n upon depth z. To either

side of the inhomogeneity (surface transition) the space supports

plane wave propagation. However, for a metallic medium the

refractive index is imaginary, or complex if losses are present,

and the transmitted fields are evanescent.
originates in the z < homogeneous region (e.g., vacuum), and impinges

upon the transition at z = with incidence angle e-| . The incident fields

propagate with diminishing intensity through the transition to emerge in

the uniform interior (z > 0) as a transmitted plane wave. The reflected

energy is visualized as building up, from zero energy in the uniform

interior to full value at the surface z = 0, as the fields are traced

through the inhomogeneity toward the source.

2.1 Reflection Coefficient for an s-Polarized Plane Wave

The exterior, incident and reflected field directions for an s-polarized

plane wave are shown in figure 2a. For a medium which varies only with

depth, the interior field configuration sufficiently below the surface

must be planar and must have the same polarization as the exterior fields.

Within the surface transition the fields are not planar and here is

where the reflection develops. Throughout the medium, all fields are

assumed to vary with time as exp(-io>t), the electric vector is oriented

in the y direction perpendicular to the plane of incidence (s-polarization)

and the corresponding magnetic vector has only x and z components. Due

to the symmetry of the problem the fields are independent of y and Maxwell's

equations are simply:

8E..( x,z) o l \ \

= -iu3u(z)H

x (x,z), (2M)

9E v (x,z) (?-?\ Y

, = + ia>y(z)H_ x,z), u d)

3x Z ' '

9H

x (x,z) 3H

z (x,z) . (2-3)

-hi hr~ = -^z Vx ' z) -
z

(a) Reflection of an s-Polarized Plane Wave

(b) Reflection of a p-Polarized Plane Wave

Figure 2. Field directions for the two principle polarizations of an incident plane wave. The exterior fields are always planar;

the interior fields become planar at depths beyond the surface

transition.
Substitution of eqs. (2-1) and (2-2) for H x and H

z into eq. (2-3)

gives the following equation for the electric field:

r iL +

3? 1 Mil L_ + k„V(z)l F(x7)-Q (2" 4 ^ L 3x

2 +

9z 2 UTIl dz 3z

+ ko Mz)j Ey

(x,z) - ,

where n(z) = /y(z)e(z)//y e is the refractive index and

k = w/y e is the propagation constant of the wave in vacuum.

Examination of eq. (2-4), which for a homogeneous region reduces to a

Helmholtz equation, suggests the trial solution

Ey

(x,z) = E(z) exp [ik n(z) sin e(z) x] . (2-5)

Within a homogeneous region E, n and 9 attain constant values and

specifies the direction of plane-wave propagation relative to the z axis.

When trial solution (2-5) is substituted into eq. (2-4) and similar

terms are grouped together the following expression is obtained after

canceling a common exponential factor:

|[g-^l + ^(l-si,A,E

] +

[2(^| - E 1 ^)ij_ (nsinQ) + E ^ (nsinQ) ] (ik x) +

eT^ ( nsin e)]2 (ik x)2 } = •

< 2

- 6 > L dz J ° \

Each term within this expression has been factored into a function only

of z (contained by a square bracket) and a power of (ik x). Unless the zdependent factor of each individual term is made to vanish, the sum of
terms in eq. (2-6) can not be zero for all values of (ik x). Setting

the coefficient of (ik Q

x) 2 to zero immediately provides, for E * 0,

d (n sin e) = 0> or n(z) sin e(z) = constant.

cTz (2-7)

This relation also satisfies the requirement that the coefficient of (ik x)

must vanish. The constant appearing in eq. (2-7) is evaluated by means of

the boundary condition that within the homogeneous region containing the

incident plane wave the angle of incidence is e-j and the refractive

index has the constant value n-| . This results in a generalization of Snel

1

'

s

law of refraction for a continuous z-dependent medium:^

n(z)sin e(z) = ni sin 8-| . (2-8)

The present significance of eq. (2-8) is that the exponent of the assumed

solution, eq. (2-5), is now seen to depend only upon x.

The remaining term in eq. (2-6) yields an equation for the z-dependent

electric field amplitude, cf. eq. (2-4),

£l - 1 du dE + K 2

E

= , (2-9)

<j z

2 u dz dz

where

3 A physical argument supporting eq. (2-8) can be made by dividing the

medium into a large number of homogeneous slabs. At each interface

Snel l

1 s law applies so that n -| s i n -| = ri2sin62 = n3Sin93 = . . . = n-jSinS.,-. In passing to the limit of infinitely many slabs of vanishing

thickness, eq. (2-8) is recovered.
< = <(z) = k n(z) cos 6(z) = k o y n

2 (z)-n-| 2 sin2

6i

(2-10)

It will prove convenient to divide E(z) into two parts which in the initial

homogeneous region can be interpreted as the amplitudes of the incident

and reflected fields, respectively,

E(z) = Ei(z) + E r (z), (2-11)

and furthermore to define

$, = ik(z)[E,(z) - E (z)]. (2-12)

dz ' r

The effect of replacing the two functions E(z) and dE(z)/dz by two

new functions E-j(z) and E r (z) is two-fold. Physically , the electric and

magnetic field components now resemble their counterparts in a homogeneous

medium. According to eqs. (2-5), (2-8), and (2-11)

E (x,z) = [E^z ) + E r (z)]exp[ik

Q

n

1 sin e

] x] , (2-13)

while from eqs. (2-1), (2-2), and (2-13), in view of eqs. (2-7), (2-8),

(2-11), and (2-12):

H

x (x,z) =JiLz) [-E.(z) + E r (z)]cos efztexpCik^sin e

] x] , {2 _ 1 4)

H

z (x,z) =JI[^ lE.(z) + E r (z)]sin e( z )exp[ik

() n

1 sin e

] x] . (2 _ 15)

10
Mathematically , the effect is to reduce the single second order differential

equation (2-9) to two first order differential equations. One equation

is obtained by substituting (2-11) into (2-12):

5i + 5r-i<(E,. -EJ =0 (2-16)

dz dz l r

The other equation is obtained by eliminating d^E/dz^ from eq. (2-9) using the

derivative of eq . (2-12) and then substituting eq. (2-11) for E(z).

|L.|r.,.c(E1 + E

r

) +2r

s

( El -E

p

> =0, «-">

where

r = 1

"• d 1 = 1 d In JS_ (2-18)

s "2~ </\i cfz" y 2 cTz uy

Now, half the difference of eqs. (2-16) and (2-17) is

^ + i<E rt r s(Er . Ei ,=o «-!»>

whereas half the sum of eqs. (2-16) and (2-17) is

^l-lKEf -r

$ «Er .E1 )-0 <2"2°>

Multiplying eq. (2-19) by E-j , eq. (2-20) by E r

, and subtracting one

equation from the other, gives

11
E .^T. E ^1 + 2 i<E.E -r (E. 2 -E 2

) = • (2-2D

Dividing eq. (2-21) by E.j , and introducing the definition

EJz)

Rc(z) r

v

k s E^z) (2-22)

allows eq. (2-21) to be written in the form:

^-s + i2K(z)R c - r ( Z )[l-R 2] = o

• (2 -23)

dz s s s

Equation (2-23) is a Ricatti-type ordinary differential equation [11];

it is of first order and non-linear in the dependent variable R s (z).

The following physical interpretation is offered for the function R s (z):

Within the z < homogeneous region both k(z) and y(z) are constants

and eq. (2-9) reduces to a simple Helmholtz equation. Consequently

E-j(z) and E r (z) can be identified here, respectively, as incident and

reflected plane-wave field amplitudes. Furthermore, in accordance with

definition (2-18), r

s vanishes and eq. (2-23) is readily integrated

to give

R

s (z) = Rs (0)exp (i'2kz) = Rs (0)exp (i2k niCOs8-|z) . (2-24)

The function R s (z) defined by equation (2-22) for z < therefore has the

properties required of a reflection coefficient.

12
Deep within the medium, below the transition layer, where the

homogeneous properties again prevail <(z) is again a constant and

eq. (2-9) again yields plane wave solutions. However, because neither

sources nor boundaries appear in this region, E r (z) must vanish here

and E-j(z) must assume the nature of the transmitted plane wave field.

As a consequence, the appropriate boundary condition for differential

equation (2-23) is that R

s (z) = at z = °°, i.e., at depths well

within the uniform interior.

Within the transition layer the quantity r

s (which is a measure

of material inhomogeneity) is no longer zero in eq . (2-23) for R

s (z),

while <(z) in eq. (2-9) looses its identity as a plane wave propagation

constant. This is evident for a gradual transition (small r

s

) in a

medium of uniform permeability. In that case the WKB approximation [12]

should hold and the solution of eq. (2-9) takes the form:

E(z) = E^z) + E r(z) = A(z)e i<(z)z + B(z)e- iK ( z ) z

. (2-25)

A Fourier spatial decomposition of either term would reveal an infinite

spectrum of plane wave components traveling in both positive and negative z

directions. That is to say, neither E -,- ( z ) nor E r (z) has a totally unique

velocity of propagation and may even contain stationary wave components.

A closer inspection of eqs. (2-19) and (2-20) reveals, in a more

general situation, that within the transition layer E-j(z) and E r (z) are

inseparably coupled by the inhomogeneity (r

s

) of the medium. If E-j(z)

and E r (z) were truly normal modes of the system (i.e., planar field

amplitudes) they would surely satisfy independent equations. Dividing

E(z) into components E-j (z) and E r (z) is merely a convenience, without

13
loss of mathematical rigor, which leads to a physical understanding of

the fields whenever the medium is homogeneous and r

s (z) vanishes.

In view of what has been said it might appear that no physical

interpretation can be attached to the ratio R s (z) = E r (z)/Ej(z)

within the transition. That is not so. Consider the- impl i cations of a

numerical integration of eq. (2-23) via a step-by-step procedure starting

deep within the medium at large positive z, where R s = 0, and progressing

through the transition layer to the surface at z = 0. The current value

of Rs as the numerical integration reaches the surface is indisputably

the plane wave reflection coefficient for the material half-space. In

the same manner, if the integration were halted at some depth prior to

reaching the actual surface, the then current value of R

s should logically

be identified with the plane wave reflection coefficient for the given

inhomogeneous medium when terminated at this depth by a uniform medium of

matching electrical properties. A consequence of this isomorphism, which

holds for all z, between the function R s (z) and the plane wave reflection

coefficient is that:

|R s (z)| < 1 for all z . (2-26)

This is true because, by energy conservation principles, the plane wave

reflection coefficient is prevented from exceeding unit magnitude.

2.2 Reflection Coefficient for a p-Polarized Plane Wave

The p-polarized plane wave is configured as in figure 2b with the

magnetic vector in the y direction, perpendicular to the plane of incidence,

The electric vector is oriented parallel to this plane and has only x

14
and z components. In a medium which changes only with depth z the

electromagnetic field excited by the plane wave must be independent of

the y coordinate and consequently Maxwell's equations for an assumed

exp(-iwt) time dependence and this polarization reduce to:

3H

y

(x,z) -JTT-

= +lwe(z)E

x( x > z > • (2 _ 27)

aH

y

(x ' z) "J • i \c ( \ Tx—

= -i-(^2 (x,z)

. (2 .28)

3E

x (x,z) 3E

z (x,z)

= +iwy(z)H..(x,z)

dZ 3X

.«H»-.f. r

«,-i • (2 _ 2g)

A comparison of these three equations with eqs. (2-1), (2-2), and (2-3)

•* ->->• reveals that if in the latter set of equations E is replaced by H, H is

- replaced by -E, and the material parameters e(z) and u(z) are interchanged then the above set of equations is obtained. By means of this

duality the differential equation for the p-polarized plane wave reflection

coefficient can be written down from inspection of eq. (2-23). Namely,

^P + i 2 <(z)R - r ( Z )[l-R n

2

] = , (2 " 30)

dz P P P

where now

R (z) = ^ < 2

~ 31 ) P U)

H^z)

is the reflection coefficient for the magnetic field. According to eq.

15
(2-10) k(z) is unchanged by the transformation, whereas according to eq

(2-18):

r=l 1 l_i=ld_lni_ . (2- 32 ) P 2 <7e" dz e 2 dz ooe

As in eq. (2-26) the magnitude of R

p (z) obeys the condition:

R

p

| <_ 1 for all z . (2-33)

At normal incidence the reflection coefficients for the s- and ppolarized waves must be identical because the plane of incidence

degenerates into a line of axial symmetry. This is easily verified

because, for 9] = 0, < = go/mF and

r = 1 fe^u . u de\ = . r . (2-34)

s ^F \ dz dz J P

Writing r for r

s or -r

p

, and R for R s or -R

p

, eqs. (2-23) and

(2-30) can be put into a common form for the normally incident plane

wave

^R + i2<(z)R - r(z)[l - R

2

] =

(2 " 35)

dz

At normal incidence R

p

= H r /Hj is the negative of R s = E r

/E-j , a result of

the field directions assigned in figures 2a and 2b.

16
2.3 Singularities of the Reflection Coefficient Formulation for a Lossless

Jellium Metal

For either polarization of the incident plane wave fields the

reflection coefficient is determined by solving the following differential

equation for R-j(z) ,

dR—i + i2k(z)R

i - r.( z ) [1-Rj 2

] =0, i = s or p , (2 ' 36)

cf. eqs. (2-23) and (2-30), subjecting the solution to an appropriate

boundary condition, such as Rj(») = 0, and then evaluating the solution

at the surface of the medium. For a metal described as a lossless, nonmagnetic, depth-dependent jellium the permeability and permittivity

functions [13] are simply:

viz) = y (2-37)

and

e(z) = e [l - (u> p

/wK] . (2-38)

In eq. (2-38) the plasma frequency u

p is a function of depth z through

its dependence on the local electron density. The coefficients of differential equation (2-36) evaluated by means of eqs. (2-10), (2-18) and

(2-32) for an incident wave originating in vacuum, where n-| = 1 , are

explicitly given by:

17
(z) = - yw2 cos

2

i 1 - w

p

(z) , (2-39)

r

s (z) = -}[w

2 cos2

e

1 - ^(z)]"1 ^U 2

) , (2-40)

r n (z) = + ll>2-u> ^(z)]"1 4rU 2) , dz VU) p (2-41)

where c = l//y e is the vacuum speed of light.

Differential eq. (2-36) is clearly singular at the pole of r^ . For

the s-polarized wave this occurs at a depth of z

s satisfying the relation

Wp(z s

) = wcos6]; for the p-polarized wave this occurs at a depth of z

p satisfying the relation w

p (zp

) = w. In order to examine the behavior of

R-j at a pole of r^ , eq. (2-36) is divided through by r^ and r^ is allowed

to pass to infinity. At the pole of r^ it is assumed that R-j is welldR,

behaved and continuous, and —-1 remains finite. Under these conditions

dz

equation (2-36) reduces to

R.j(z.j) = 1 , i = s or p (2-42)

Total reflection is seen to occur at a plane z-j where r^ is singular.

Examination of the generalized law of refraction (2-8) for n-| = 1

and n(z) = *e/e = A - (w /a>) 2 reveals, in the plane z = z

s where

where ojp

(z) = tocos e-j , that 6(z s

) = */2 for either polarization of the

field. In other words, the rays of the optical field are turned about

at the same plane, z = z

s

, at which the s-polarized wave is totally

reflected. The turning plane of the optical field does not, however,

coincide with the reflection plane of the p-polarized wave for reasons

that will be made clear later. In any case, for either polarization

18
of the wave and for z < z

s

, 6(z) is real, <(z) is real and the fields

given by eq. (2-9) or its dual are oscillatory. For z > z s the law of

refraction (2-8) gives an imaginary value for e(z) and eq. (2-39) shows

that k(z) is imaginary. The fields of either wave polarization exibit

an exponential character of decay in this region.

The p-polarized wave has an electric field component in the z

direction (not present in the s-polarized wave) which, though rapidly

attenuated with depth below the turning plane z s , is capable of exciting

a non-propagating resonance^ of the jellium plasma at a depth z > z_.

At this plane, where e(z

p

) = 0, Maxwell's equations (2-26) through (2-28)

require the magnetic field to vanish and if an electric field exists that

it must be expressed as the gradient of a scalar function. The electric

field is in fact an electrostatic oscillation of the medium at the local

plasma frequency ^

p (zp

) = w. Since the plasma resonance at z

p draws

energy from the turning plane of the optical field at z

s the location

where R

s 2 = 1 differs from the location where R

2 = 1.

It is useful to know whether R-j(z-j) = +1 or -1 so that R-j(z-j) can

serve as an alternative to the boundary condition R-j(°°) = 0. This is

achieved by examining the field expressions in their respective reflection

planes. Since Hy(x,z) for the p-polarized wave is given by an equation

dual to eq. (2-13) for the s-polarized wave, viz.,

H (x,y) = [H

r

(z) + H.(z)] expdk^sin e^) (2-43)

4 In the appendix it is proven that an obliquely incident wave must have

an electric field component in the direction of the material gradient in

order to excite a plasma oscillation.

19
the vanishing of Hy

in the plane zp

of the plasma resonance leads to

the boundary condition

W = rfjf} = -1 • (2-44)

For the s-polarized wave consider the possibility of Rs (zs) = _1 •

This implies E r (z s

) = -E-j (

z

s

) and, since k(z s

) = 0, according to eqs.

dE

(2-11) and (2-12), that both E and ^i" vanish at z

s

. Equation (2-9) and

its derivatives further show that all higher derivatives of E with respect

to z must also vanish at z

s

. According to Taylor's theorem this means

E(z) = for all z and so there can be no electromagnetic field anywhere

within the medium. Since this contradicts experience, R s (z s

) * -1.

The alternative possibility, R

s (z s

) = +1, leads via eq. (2-14) to

H

x (x,z s

) = but, by eqs. (2-13) and (2-15) to non-zero Ey

(x,z s

) and

H z (x,z s

) field components. The Poynting's vector (E * H*) for these

components points, as it should, in the positive x direction, i.e.,

6(z s

) = tt/2. The boundary condition for the s-polarized wave is

therefore:

R

s (z s

) = +1 (2-45)

Singularities of the reflection coefficient formulation of the type

encountered with the present lossless-jellium model, can sometimes be

avoided by working with the wave immittance formulations described next.

A bilinear algebraic transformation links the wave immittance function

to the reflection coefficient.

20
3. WAVE IMMITTANCE FORMULATIONS FOR AN ARBITRARY ONE-DIMENSIONAL MATERIAL

TRANSITION

The preceding theory illustrates the relationship which exists

between Maxwell's (curl) equations for the fields, the wave equation,

and the ordinary (but non-linear) differential equation for the reflection

coefficients. The wave impedance and wave admittance formulations offer

still another avenue for describing plane wave reflection phenomena.

The differential equations governing the wave immittances and the relationship which exists between the immittance and reflection functions are

derived next.

3.1 Wave Immittance Formulations for s-Polarized Plane Wave Reflections

The wave impedance, Z

s (z), at some given plane z is defined as the

ratio of electric- to-magnetic field components which in that plane leads

to positive z propagation. For an incident s-polarized plane wave this

ratio is equal to

Z(zM s -H^4x (x,z)

(3-1)

From eqs. (2-13) and (2-14) it is evident that Z

s is independent of x

and a function only of coordinate z. The derivative of this function.

dZ

s (z) -1 aE y

E v 3H X dz H x 9z H

2 3z

(3-2)

is easily simplified using Maxwell's eqs. (2-1) and (2-3) to eliminate

the field derivatives on the right:

21
dz

s (z) • / > r • <\ 1 aHz1/ E v\

2

(3-3) _|_.,«M(z) + [.,»e{2)-t^J^ . (33)

The x derivative of eq. (2-2), together with definition (3-1), permits this

expression to be rewritten as

^Si£] . 1uM(z , >\-Ueiz) + ' ^£^lz52 (z) . '(3-4)

dz L lup(z) E 3x2 J

b

The remaining field derivative is readily eliminated as a consequence of

the exponential x dependence of Ey appearing in eq. (2-5) so that

dZ

s — (z) . §— = i^(z) + dz

(3-5)

, L, [g»2 M ( z )e(z) - k V(z)sin2e(z)]Z 2(z) . i wii ( z

)

° s

Introducing the relation k Q

2 n2 (z) = oj 2 u(z)e(z) leads to the following

following differential equation for Z s (z):

^|ii! - i W(2)[ l.z

s 2(z)/Z S0

2 (z)] - , (3 " 6 >

where Z s0 (z), the nominal impedance of the medium at z, is defined by

Z (z) = jMz) J = o)m(z) (3-7)

so

12

' " \7TzT7oTeT71 TTTT

and k(z) is given by eq. (2-10).

22
The wave admittance, Y(s), is the reciprocal of the wave impedance.

The differential equation for the wave admittance is most easily derived

by placing

Z

s (z) = 1/Y s (z) and Z so (z) = l/Y S0 (z) 3-8)

in eq. (3-6). This immediately yields the following result:

^^ -i<(z)Y SQ (z)[l - Y

s 2 (z)/Yso

2 (z)] = (3-9)

Observe that differential eqs. (3-6) and (3-9) for the wave immittances

are both Ricatti-type equations. For a particular medium the location of

the poles and zeros of the immittance functions will determine whether it

is more convenient to use eq. (3-6) or eq. (3-9).

In the theory that has been developed for uniform transmission

lines, waveguides of constant cross-section, and plane waves in homogeneous

media, the reflection coefficient is related to the wave impedance via a

bilinear transformation such as:

„ (z)

- V*) - z SQ (z)

s

' Z

s (z) + Z so (z)

(3-10)

where, for the cases mentioned, Z so is a constant. It is tempting to

postulate that such a relation holds for the present case even though Z s0

is not a constant. The validity of the bilinear transformation for the

present situation is demonstrated by substituting eq. (3-10), with a

z-dependent Z s0 , into eq. (2-23) for R

s

; this yields:

23
dz L-SO

.so + 2 r. Zc - IKZcnCl - (Zc/Zc n ) 2 s^so ] = (3-11)

The second term in this equation vanishes because of eqs. (3-7) and (2-18)

and <Z S0 = up. Consequently eq. (3-6) is recovered and the applicability

of transformation (3-10) for s-polarized plane wave propagation in depthdependent media is established. The corresponding transformation for the

admittance function is obtained by substituting eq. (3-8) into eq. (3-10).

This yields in a similar manner:

R s (z) = - Vz) - Y so (z)

Y

s (z) + Y so

(z)

(3-12)

Below the surface transition, within the uniform volume of the

dZ<. dY<. medium, lAi) and Y<.(z) are constants. Consequently both —* and

i

b b dz dz

must vanish here, and eqs. (3-6) and (3-9) show that Z s (z) and Y

s (z)

respectively approach the constant values assumed here by Z s0 and Y s0 .

The boundary conditions for eqs. (3-6) and (3-9) are therefore explicitly:

Ze(-) =

cop Zso(°°) = — (3-13)

and

v^(oo) = y^^(°°) = — 's v

' 'so v

' u)p (3-14)

The equivalence of these boundary conditions with the boundary

condition R

s

( DO

) = is readily demonstrated by substitution of

eq. (3-13) into eq . (3-10) evaluated at z •> °°, and substitution of

eq. (3-14) into eq. (3-12) evaluated at z

»• °°.

24
3.2 Wave Immittance Formulations for p-Polarized Plane Wave Reflections

The equation obeyed by the admittance Y

p (z) of a p-polarized plane

wave may be found by differentiating the definition

Y (z) = Mllll

V Ex

(x,z) ' (3-15)

or by invoking the duality which exists between this function and the

impedance function defined by eq. (3-1) for the s-polarized plane wave.

The latter course is the more expedient and requires the exchange of

y with e, Z s with Y

p

, Z so with Y p0 in eqs. (3-6) and (3-7). This

leads to the following Ricatti differential equation for the wave admittance:

&i_ - io)£( z )[l - Y

2 (z)/Y 2 (z)l = dz P lz;/T

po

KZU u

' (3-16)

where Y p0 (z), the nominal admittance of the medium, is given by:

Y (z) sJs(z) 1 = u>e(z)

po yVTDVTzl cose(z) ~^7T Zp0 (z) (3-17)

The wave impedance satisfies the dual of eq. (5-9), namely,

¥sj±} - i«(z)Zn .(z)[l - Z

n

2 (z)/Z nn2 (z)] = (3 - 18)

dz P° P P°

the reflection coefficient is likewise related to the immittance functions

by means of bilinear transformations dual to eqs. (3-10) and (3-12):

r ( 7 ) -

Y

P (z) ~ Y PQ (z) - Zp (z) - Z p0 (z) (3 . 19)

%

[Z)

' Y

p (z) + Y p0 (z) " " Zp (z) + Z p0 (z)

25
The boundary conditions, dual to eqs. (3-13) and (3-14), for

differential eqs. (3-16) and (3-18) are, respectively:

toe YT (oo) = v ~(°°) = — p

v ' 'po v

' K (3-20)

and

Z

p («) = Zp (-) =

<

oTe (3-21)

These are, of course, equivalent to the boundary condition R

p

(°°) = 0, as

readily demonstrated using eq . (3-19).

3.3 Singularities of the Wave Immittance Formulations for a Lossless

Jell i urn Metal

The wave immittance formulations are now specialized to a lossless,

non-magnetic, depth-dependent jellium metal. Once again eqs. (2-37),

(2-38), and (2-39) apply, and for the s-polarized wave eqs. (3-7) and

(3-8) give

Z so (z) = Wi 2 w

d 2(z ) -T y so

(z)

(3-22)

Substitution of this expression together with the definition

r o %2{z)1 F

s (z) s i^ [cos2 e

1 -—

E-^2—

J

(3-23)

into eqs. (3-6) and (3-9) yields the following differential equations for

the s-polarized wave immittance functions:

26
2s + F (z)Z 2

= 1«V (3 ~ 24)

and —5 + i^n

Y

2

= F c (z) * (3 " 25)

dz ° s s

For the p-polarized wave, eq. (5-17) yields

2/„\_ I I ~

2777 (3-26) v-/^?]/fs

2 fil V' z)_

Substitution of this expression along with the definition

r ? un (z) 1 / r w n (z) n F

p

(2

' =

i -oh e i--V]/ C1 -^] (3-27)

into eqs. (3-16) and (3-19) provides the following differential equations

for the p-polarized wave immittance functions:

£L? + F (z)Y 2 = 1»e(z) . (3 " 28)

dz V P

v ^ ;

and

^P + 1«oe(z)Z 2

= F (z) . (3 " 29)

dz P P

Differential equations (3-24), (3-25), (3-28), and (3-29) for the

immittance functions are generally more tractable Ricatti equations than

27
those of (2-36) for the reflection coefficient. As a result of transforming

to the immittance formulation, the singularity at u) p (z s

) = wcose that

plagued the differential equation for R s (z) no longer appears in either

eqs. (3-24) or (3-25). The choice between the wave impedance formulation

or the wave admittance formulation is straight-forward. For the spolarization the admittance formulation is preferred because the coefficients

on the left of eq. (3-25) are constants. For the p-polarization the

impedance formulation is selected to avoid having the plasma resonance

pole of F(p) appear as a coefficient of Y 2 (z).

The boundary conditions for the immittance functions at infinity, as

expressed by eqs. (3-13), (3-14), (3-20), and (3-21), remain appropriate

for use when, for example, Y so («) and Y p0 (°°) are evaluated by means of

eqs. (3-22) and (3-26). However, it may at times be more convenient

to employ the boundary conditions for the immittance functions at the

reflection plane. To obtain these conditions the bilinear transformation

of (3-10) is inverted to give

z (z ) =

] + R(

fiLl z (z ) (3-30) ZsUs)

1 - R(z

s

) Z ° sUs)

At the reflection plane z

s of the s-polarized wave, ^

p (z s

) = wcos9-| .

Here the nominal impedance of the medium given by eq. (3-22) is infinite,

but it is also true that R s (z s

) = +1 according to eq. (2-45). Thus the

boundary conditions on the s-polarized wave immittance functions are:

Z s (z s) = - or Y

s (z s

) = (3-31)

28
The inverse bilinear transformation for the p-polarized wave may be

determined from eq . (3-19) but it is also the dual to eq . (3-30), viz.,

Y (z ) ,

] + W y ( Z , P P 1 - R (z ) Y P° UP

; (3-32)

At the reflection plane Zn , wp(zp) = w. Here the nominal admittance given

by eq. (3-26) is zero, but it is also true that R

p (zp

) = -1 according to

eq. (2-44). It therefore follows that the boundary conditions as the ppolarized wave immittance functions are

Y p (z

p

) = or Zp(Zp) = (3-33)

4. CONCLUSIONS

The reflection of an obliquely incident, monochromatic plane wave

from the surface of a graded medium has been investigated. Several

alternative, but related, formulations were presented in order to provide

flexibility in solving a particular problem. Among the advantages

of the present formulations are that Maxwell's equations do not need to

be solved for the field configurations, the theory is exact, and its

application is straightforward. The selection of a particular formulation

will, in every case, be guided by the characteristics and the depthdependence of the medium.

5. REFERENCES

[1] Foersterling, K. Lichtfortpflanzung in inhomogenen medien. (Theorie

der Lippmannschen farben-photographie) . (I) Physikal ische Zeitschrift

14: 265-270; 1913. (II) Physikal ische Zeitschrift 15: 225-234; 1914.

29
[2] Born, M.; Wolf E. Principles of optics. New York: Pergamon Press;

1980. 280.

[3] Foersterling, K. Ueber die ausbreitung des lichtes in inhomogenen

medien. Annalen der Physik Y\_: 1-39; 1931.

[4] Geffcken, W. Reflexion elektromagnetisher wellen an einer inhomogenen

schicht. Annalen der Physik 40; 385-392; 1941.

[5] Kofink, W. Reflexion elektromagnetischer wellen an einer inhomogenen

schicht. Annalen der Physik J_: 119-132; 1947.

[6] Brekhovskikh, L. M. Reflexion of plane waves from layered inhomogeneous

media. Zhurnal Teknischeskoi Fiziki 19; 1126-1135; 1949.

[7] Bailey, V. A. Reflection of waves by an inhomogeneous medium.

Physical Review 96: 865-868; 1954.

[8] Pierce, J. R. A note on the transmission line equation in terms of

impedance. Bell System Technical Journal 22: 263-265; 1943.

[9] Walker, L. R.; Wax, M. Non-uniform transmission lines and reflection

coefficients. Journal of Applied Physics V7_: 1043-1045; 1946.

[10] Tolstoy, I. Note on the propagation of normal modes in inhomogeneous

media. Journal of the Acoustical Society of America 27^: 274-277;

1955.

[11] Davis, H. T. Introduction to nonlinear differential equations.

Atomic Energy Comm. (U.S.) Govt. Printing Office; 1960. 57-78.

[12] Morse, P. M.; Feshbach, H. Methods of theoretical physics. New York:

McGraw-Hill Book Co.; 1953. 1092-1106.

[13] Kittel, C. Introduction to solid state physics. New York: John

Wiley and Sons; 1966. 227-229, 244.

30
APPENDIX. INCIDENT FIELD POLARIZATION FOR EXCITING A PLASMA RESONANCE

For an electromagnetic field oscillating according to an exp(-iwt)

time dependence, the magnetic induction H satisfies the Maxwell equation

VxH = -iwD (A-l)

The field displacement, D, is related to the electric intensity, E, through

D = e(w)E (A-2)

In writing this expression the conduction current has been lumped together

with the displacement current to form the complex permittivity function

s(w ). Since the divergence of the curl of any vector must identically

vanish, eq. (A-l) leads to

V»D = (A-3)

or, using eq. (A-2) , to

e V»E + E'Ve = (A-4)

For a homogeneous medium Ve = 0, leaving v»E = 0. Both the bound charge

density Pb = v»D and the free charge density p s = v«E are consequently

zero within a homogeneous medium.

For a graded medium the bound charge density remains zero,

P b

= V«D = (A-5)

31
However, the free charge density,

p = ve =

- E' Ve = -1 E. de (A-6

z "ar

resulting from a displacement of the electronic charge from its equilibrium

position, differs from zero if E z

* 0. Only the p-polarized wave has

such an electric field component, and therefore only a wave of this

polarization is capable of exciting an oscillation of the free charge.

The amplitude of these oscillations increases as the resonance condition

e = is approached. If collision losses are ignored the permittivity

function for a jellium model is given by

e = e [l-<y/(z)/</] . (A-7)

According to eq . (A-6) resonance is seen to occur at a depth and plasma

frequency satisfying the condition e = 0, i.e.,

wp(z) = a) . (A-8)

32

•& U.S. GOVERNMENT PRINTING OFFICE : 1983 O—380-997 (2397)
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS TN 1171

2. Performing Organ. Report No. 3. Publication Date December 1982

4. TITLE AND SUBTITLE

An Electromagnetic Formulation for Treating Optical Reflections from Graded-Material

Surfaces

5. AUTHOR(S)

A. George Lieberman

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Same as item 6.

10. SUPPLEMENTARY NOTES

J

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) The reflection of a monochromatic plane wave falling obliquely upon the surface of

an arbitrary, flat, depth-dependent material is investigated theoretically. The

complex reflection coefficient for either principal (s or p) polarization of the

field is shown to satisfy a non-linear differential equation of the Ricatti type.

An alternate formulation based on the wave immittance (i.e., impedance or admittance ) functions is also presented. The immittance functions are shown to satisfy Ricatti

differential equations of their own. The reflection coefficient formulation and the wave immittance formulations are related via a bilinear algebraic transformation.

Singularities appearing in the reflection coefficient formulation may be suppressed

in the immittance formulation, and vice-versa. The advantage of either formulation

is that the reflection coefficients for an arbitrary, depth-dependent medium can be

obtained directly without having to solve Maxwell's equations for the internal field

configurations.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separote key words by semicolons)

electromagnetic waves; graded materials; inhomogeneous media; jelnum;

optical reflections; reflection coefficient; Ricatti equation; surface reflections;

wave immittance.

13. AVAILABILITY

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14. NO. OF

PRINTED PAGES

36

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